Partial Fractions Partial Factions In algebra we learn how to find the common denominator of two expressions and then combine like terms.  Partial fractions are all about going backwards.   Example 1 Consider the rational function                        3x + 2                      3x + 2         P(x)  =                        =                                                        x2 - 1                   (x - 1)(x + 1) We want to write it in the form                 3x + 2                        A                B                                         =                   +                             (x - 1)(x + 1)                 x - 1           x + 1   To do this we need to solve for A and B.  Multiplying by the common denominator         (x - 1)(x + 1)  we have         3x + 2  =  A(x + 1) + B(x - 1)   Now let x  =  1         5  =  2A + 0         A  =  5/2 Now let x  =  -1         -1  =  -2B         B  =  1/2 Hence we can write              3x + 2                    5/2              1/2                                  =                   +                                x2 - 1                   x - 1             x + 1   This is called the partial fraction decomposition of P(x) Example 2 Find the Partial Fraction Decomposition of                        3x2 + 24x + 7                 3x2 + 24x + 7         P(x)  =                                 =                                                           x3 - 2x2 + x                        x (x - 1)2   Since we have a square, we need to write the decomposition in the form below.              3x2 + 4x + 7                 A                 B              C                                        =                 +                   +                                                     x (x - 1)2                  x - 1          (x - 1)2          x   Multiplying by the common denominator, we have         Ax(x - 1) + Bx + C(x - 1)2  = 3x2 + 4x + 7 Let x = 0:         C = 7 Let x = 1: We have          B = 14 Now look at the highest degree coefficient:         Ax2 + Cx2  = 3x2 Dividing by x2 and substituting C = 7         A + 7 = 3,     A = -4 We conclude that             3x2 + 4x + 7                 -4                14               7                                       =                  +                   +                                                    x (x - 1)2                  x - 1           (x - 1)2          x Integration Example Evaluate the integral           Solution First we find the partial fraction decomposition of the integrand.  We factor to get                          4x + 2                      A                B                                      =                  +                           (x - 2)(x + 3)               x - 2           x + 3 Multiply by the common denominator to get         A(x + 3) + B(x - 2)  =  4x + 2 Let  x  =  2:         5A  =  10        A  =  2 Let  x  =  -3:         -5B  =  -10        B  =  2 Now we can integrate         Logistics Growth It has been said that the total population of South Lake Tahoe should never exceed 30,000 people.  If in 1980 the population was 15,000 and in 2000 it was 20,000, when will the population reach 25,000.     Solution:   We make the assumption that the rate of increase of the population is proportional to the product of the current population and 30,000 minus the current population. That is:            dP                     =  kP(30,000 - P)            dt This is a separable differential equation.  Separating gives                      dP                                                             =   k dt             P(30,000 - P) Now integrate both sides.  The integral of the right hand side is          kt + C To integrate the left hand side, use partial fractions:                        1                      A               B                                         =           +                                   P(30,000 - P)              P          30000 - P Multiplying by the common denominator:         A(30,000 - P)  +  BP  =  1         P = 0:      1  =  30,000A     or     A  =  1/30,000         P = 30,000:  1  =  30,000B     or     B  =  1/30,000 Now integrate, noticing that the denominator 30,000 - P will result in an added negative sign after substitution.  We get                1                     1                         lnP -                  ln(30,000 - P)  =  kt + C            30,000              30,000 or         =  at + b    a  =  30000k      b  =  30000C When t  =  0,   P  =  15000, so         b  =  0 When t = 20, P = 20,000, so         20a  =  ln2  or                      ln 2         a  =                                      20 hence         Finally plugging in 25,000 for P gives                       t ln 2         ln 5  =                                          20 or                  20 ln 5         t  =                   =  46.4                      ln 2 The population will reach 25,000 in the year 2026. Exercise On Monday the Ebola virus is caught by 3 people in an isolated community of 10,000 people.  By Thursday, 40 people have the virus.  How long will it be until half of the population have the virus?           Back to the Math Department Home